Six people are going to sit in a row on a bench. Romeo wasnts to sit next to Juliet. Caesar does not want to sit next to Brutus. Homer and Pierre can sit anywhere. In how many ways can these people be seated?
1. Romeo, Juliet, _ _ _ _
2. Juliet, Romeo, _ _ _ _
3. _ Romeo, Juliet _ _ _
4. _ Juliet, Romeo _ _ _
5. _ _ Romeo, Juliet _ _
6. _ _ Juliet, Romeo _ _
7. _ _ _ Romeo, Juliet _
8. _ _ _ Juliet, Romeo _
9. _ _ _ _ Romeo, Juliet
10._ _ _ _ Juliet, Romeo
There are 5*2 possibilities for Romeo and Juliet.
If we have a remaining seats with 4 in a row, (numbers 1, 2, 9 and 10 for Romeo and Juliet=4 possibilites for Romeo and Juliet), we can fill the rest
Caesar _ Brutus _
Brutus _ Caesar _
Caesar _ _ Brutus
Brutus _ _ Caesar
Each of these has two possibilities for homer and Pierre, so there are 4*4*2 = 32 possibilities for everyone when Brutus, homer, Pierre and Caesar sit 4 in a row
If three of Homer, Pierre, Brutus, and Caesar sit next to each other (numbers
3, 4, 7, 8 for Romeo and Juliet = 4 possibilities for Romeo and Juliet), we have
Caesar, Rom, Juliet, Brutus _ _
Caesar, Rom, Juliet, _ Brutus _
Caesar, Rom, Juliet, _ _ Brutus
Brutus, Rom, Juliet, Caesar _ _
Brutus, Rom, Juliet, _ Caesar _
Brutus, Rom, Juliet, _ _ Caesar
_ Rom, Juliet, Caesar _ Brutus
_ Rom, Juliet, Brutus _ Caesar
For each of these, there are 2 possibilites for Homer and Pierre
8*4*2 = 48 for these arrangements
For arrangements 5 and 6 for Romeo and Juliet, we have
Caesar _ Rom, Juliet, _ Brutus
Caesar _ Rom, Juliet, Brutus _
Brutus _ Rom, Juliet, Caesar
Brutus _ Rom, Juliet, _ Caesar
4*2*2 = 16 possibilities
Total # of possibilities is 16 + 32 + 48 = 96
I hope I did this right!!
2-39=
To find the number of ways the six people can be seated, we can break down the problem into smaller cases.
Step 1: Determine the positions of Romeo and Juliet.
Since Romeo wants to sit next to Juliet, we can treat them as a single entity. So, we have five entities: (Romeo & Juliet), Caesar, Brutus, Homer, and Pierre. Considering (Romeo & Juliet) as one, we can find the number of positions for them.
Number of ways to arrange (Romeo & Juliet) = 2
Step 2: Determine the positions of Caesar and Brutus.
Since Caesar does not want to sit next to Brutus, we have to consider the positions where they are not sitting next to each other.
Number of ways to arrange Caesar and Brutus = total ways - ways where Caesar and Brutus are sitting next to each other
Total ways to arrange Caesar and Brutus = 4! = 24
Ways where Caesar and Brutus are sitting next to each other:
Assume (Caesar, Brutus) as a single entity. We then have four entities: (Romeo & Juliet), (Caesar & Brutus), Homer, and Pierre. The number of ways to arrange these four entities is 4!.
Number of ways to arrange Caesar and Brutus sitting next to each other = 4!
Step 3: Determine the positions of Homer and Pierre.
As stated in the problem, Homer and Pierre can sit anywhere, so there are no restrictions. We can treat them as two separate entities and find the number of positions for them.
Number of ways to arrange Homer and Pierre = 2!
Step 4: Combine all the cases.
To get the total number of ways the six people can be seated, we multiply the number of ways in each case:
Total ways = Number of ways to arrange (Romeo & Juliet) * Number of ways to arrange Caesar and Brutus * Number of ways to arrange Homer and Pierre
Total ways = 2 * (4! - 4!) * 2!
Now we can calculate:
Total ways = 2 * (24 - 6) * 2
Total ways = 2 * 18 * 2
Total ways = 72
Therefore, there are 72 different ways the six people can be seated.